Download Packet #3 Expressions and Equations File

Math 7
2016-2017
Packet #3
Expressions and Equations
(Module 3)
Ms. Pricola, Mr. Unson
1
Table of Contents:
Review of Integer Rules
Vocabulary of Numbers – Natural/Whole/Integers/Rational
Review of Properties
Vocabulary (Expression / Equation / Constant/Variable/
Coefficient/Term/Like Terms/Monomial/Polynomial/
Binomial/Trinomial)
Combining Like Terms
Vocabulary (Factor / Greatest Common Factor / Factoring)
Factoring out the Greatest Common Factors (GCF)
More Combining Like Terms
Writing Expressions
Writing Equations
Adding and Subtracting Expressions
Multiple Step Word Problems
One Step Equation Solving (and Checks)
Two Step Equation Solving (and Checks)
Solving Equations by Combining Like Terms on Same Side
Solving Equations using the Distributive Property
Multiple Step Equation Solving
Multiple Step Word Problems
Representing Inequalities on number line
One Step Inequality Solving
Two Step Inequality Solving
Word Problems using Inequalities
Mixed Review of the entire Domain - Expressions and
Equations
Review for Quizzes and Tests
Review of all Domains covered so far
 Ratios and Proportional Relationships
 The Number System
 Expressions and Equations
Enrichment Section (*Intro now, continue after State
Assessment
** After State Assessment)
Topics will include:
 Solving Inequalities using the negative rule
 *Rules of Exponents
 *Square and Cube Roots
 *Scientific Notation
 *Solving Equations with Variables on Both Sides
 **Solving Simultaneous Equations Using Graphing
 **Solving Simultaneous Equations Algebraically
Page 3
Page 4
Pages 5 - 7
Page 8
Pages 10 - 11
Page 12
Pages 13 - 15
Pages 16 - 18
Pages 19 - 22
Pages 23 - 24
Pages 25 - 27
Pages 28 - 30
Page 31
Pages 32 - 33
Page 34
Page 35
Page 36 - 39
Page 40 - 42
Pages 43 - 44
Page 45
Page 46
Pages 47 - 50
Pages 51 - 58
Pages 59 - 69
Pages 70 – 74
Pages 75 – 85
Pages 86 - 98
Pages 99 –108
2
In this packet:
* Questions with this annotation were taken from the North Carolina CCSS
** Questions with this annotation were taken from the Utah CCSS
Integer Rules
Additon
1. Same Sign – Add and keep the sign of the numbers
Ex. -6 + (-4) = -10
2. Different Signs – Subtract and keep the sign of the number with the
greater absolute value
Ex. -8 + 2 = -6
-4 + 9 = 5
Subtraction You have to change the problem!!!! You must change
it to “add the opposite” then use the two rules for
addition
Ex. -8 – 5 change to -8 + -5 and your answer is -13
-5 – (-2) change to -5 + 2 and your answer is -3
6 – (-4) change to 6 + 4 and your answer is 10
Multiplication and Division – are the same two rules instead of a
multiplication sign you can have a
division sign
Rule #1 negative x negative = positive
Rule #2 negative x positive = negative
Examples: -8 x -2 = 16
(n x n = p)
(n x p = n)
-4 x 3 = -12
3
-9 ÷ -3 = 3
-8 ÷2 = -4
Natural Numbers – the set of Counting Numbers:
1, 2, 3, 4, 5,……..
Whole Numbers – the set of Natural numbers and zero:
0, 1, 2, 3, 4, 5, …….
Intergers – Positive whole numbers, negative whole numbers, and zero: ……–2, –1, 0, 1, 2, …….
Rational Numbers – Numbers that can be written as fractions.
For example, –5, 0, 12, ,
0.233333…
4
Review of Properties:
Commutative Property – the order in which numbers are added has no effect on their sum. This is also
true for multiplication; numbers may be multiplied in any order and the resulting product will be the
same.
Examples:
2+3+5= 3+5+2
4 x 5 =5 x 4
a+b=b+a
Associative Property – the way in which numbers are grouped will have no effect on their sum. This is
also true for multiplication; numbers may be grouped in any order and the resulting product will be the
same. The order of the values does not change.
Examples:
(2 + 3) + 5 = 2 + (3 + 5)
(
)
4 x (5 x 2) = (4 x 5) x 2
(
)
(a + b) + c = a + (b + c)
Distributive Property – When multiplying a number by a sum, you can multiply the number by the first
value in parenthesis, then add that product to the product of the number by the second value in
parenthesis. The resulting sum will be the same as the sum generated by adding the values in the
parenthesis first, and then multiplying by the number.
Examples:
4(2 + 3) = 4(2) + 4(3)
(
)
Zero Property of Multiplication – any number times zero has a product of zero. Also, zero times any
number has a product of zero.
Examples:
3x0=0
0 x 14 = 0
Identity Property of Addition (Additive Identity)– when zero is added to any value, the resulting sum
is that original value. Also, adding a value to zero will result in a sum of that value.
Examples:
85 + 0 = 85
0 + 93 = 93
Identity Property of Multiplication (Multiplicative Identity)– when one is multiplied by any number,
the resulting product is that number. Also, when multiplying a number by one, the resulting product is
that original number.
Examples:
24 x 1 = 24
1 x 47 = 47
Additive Inverse Property – When a number and its inverse are added, the sum is zero.
5
Examples:
5 + -5 = 0
-125 + 125 = 0
Multiplicative Inverse Property – Two numbers are said to be multiplicative inverses when they have
a product of 1.
( )
Examples:
Practice Using Properties
Name the property being used to make each line change:
1]
(
[2]
___________________
(
3]
)
___________________
)
[4]
(
)
__________________
( )
___________________
12n – 12n + 36 ___________________
36 ____________________________
_____________________________
_____________________________
5]
(
)
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
6
6] (
)
(
)
(
________________________
)
________________________
________________________
________________________
7] Rewrite using the distributive property. Do not simplify the resulting expression.
= _____________________________
8] Rewrite the following using the distributive property and then find the sums.
-4.8x + (-6.2x)
9] What is the missing addend in the following:
2.34y + _________ = 0
7
Vocabulary
Expression – a series of one or more terms, not containing an equal sign.
Examples:
4n + 3f
8 + 7g
-5y – 10 + 6k
Equation – a series of terms containing an equal sign.
Examples:
2a = 14
-50n + 10 = -40
p+8=9
Constant – a mathematical value which never changes. Examples: 3 , -5 ,
, π
Variable – a mathematical “placeholder” which may be assigned any value. Once a value is assigned,
it must remain that value for the entire problem.
Examples:
Coefficient – a constant placed before (and adjacent) a variable. The coefficient is being multiplied
with the variable. In the following example, the “3” is the coefficient:
Example: 3f
3 is the coefficient
Term – consists of either a constant, variable, or combination of both being multiplied. Terms are
separated by addition or subtraction signs.
Example: If 4 + n + 5g is the expression
then
4 , n , 5g are the terms
8
Like Terms – Those terms having the same variable and exponent. Note: the coefficients do not have
to be the same, although they may be.
Examples:
3n & 5n
-4f & 120f
Monomial – An expression with only one term. That term can be a real number, a variable, or a
product of a real number and variables with whole number exponents.
Examples:
3, x, 5xy,
Polynomial – is a monomial or the sum or difference of monomials
Binomial – A polynomial with 2 terms.
Examples: x – 3, 5x + 1,
Trinomial – A polynomial with 3 terms.
Example:
9
In each of the following, name the; terms, constants, coefficients, and like terms.
Terms
Constants
Coefficients
Like Terms
7x + 4x + 3y + 2
3x + 4 + 2x + 3
3m – 2n + 5m – 4
6 + 2x + 4x
-10x
9m + 2r – 2m + r
In each of the following, combine the like terms:
1] 8x + 5x = _____________
[2] 10y – 4y = ______________
4] 6y + 4x + 2y + 3x = ________________
6] 8x + 8y + 3x = ____________
[5] 3x + 2 + 4x + 5 = ____________________
[7] 4x – 3 + 5x = ____________
9] x – 3x + 2x + 4 = ____________________
11] 5x + 8 – 7x = ___________
[8] -5x + 8x = ___________
[10] 4x + 3 – 5x + x = _____________________
[12] -6x – 8x = ____________
14] 6x + 4 – 2x – 3 = ________________
16] -7y – 12y = __________
[3] 4 + 3x + 2 = __________
[13] 4x – x = __________
[15] 7y + 5x – 9y + 6x = ___________________
[17] 4x + 3 – 10 – 2x = ______________________
18] 2x + x + 3x – 4x = ___________________
[19] 12 – 4x + 7y – 9x – y = __________________
10
20] 6x + 12 – 30x – (-8) = _________________
[21] 8 – (-2) + 5n – (-15n) = __________________
22]
4x – 7x + 2(x + 6) + (–4) = _____________________________________
23]
–5x + 2x + 3(2x + 4y) + y = _____________________________________
24]
2(x – 3y) + 4(-3x + 2y) = _________________________________________
25]
(3x + ) – 5x + 4
= ______________________________________
11
Vocabulary
Factor – one integer is a factor of another integer if it divides that integer with a remainder of zero.
Example: 2 is a factor of 10
Prime Number – positive integer greater than one with exactly two factors 1 and itself.
Example: 13, 17, 2, 5
Composite Number – positive integer greater than one with more than two factors.
Example: 22, 36, 100
Greatest Common Factor (GCF) – the largest number that is a common factor of 2 or more
numbers
or expressions.
Example: Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
GCF is 4
Factoring – Writing a polynomial as the product of 2 factors by using the GCF of the terms as one of
the factors. Factoring is the reversing the Distributive Property.
Example: Factor 6xy + 12x
Answer: 6x(y + 2)
12
Using the Greatest Common Factor (GCF)
Find the GCF of the following:
A] 3a + 4a ; gcf = _________
B] 12b + 4c ; gcf = ________________
C] 10 – 2c + 4d ; gcf = _____________
D] d + 4d + 5d ; gcf = ________________
E] 6f + 12c – 3 + 18d ; gcf = _________
F] 4w – 6w + 10w ; gcf = __________
Rewrite each of the following using the GCF as a product of two factors:
1. 15a + 5 = _________________
2. 16d – 6 = ________________
3. 20k + 14m – 2 = _______________
4. 18h – 8f + 4 = ___________________
5. 20 – 10y = _________________
6. 18 + 9c + 27a = ____________________
7. 4w + 12k – 10p = _______________
8. 17r + 51a = __________________
9. -7p + 14g = ________________
10. -7p – 14g = _________________
13
11. -6w + 10r – 4 = _______________
12. -6w – 10r + 4 = ____________________
13. Jerry has 4 jelly beans, Suzi has 12, and Billy 8. What is the largest number of jelly beans they
could each use to make equal numbered piles? (All 3 kids must have the same number in each pile)
Factoring Out GCF
1. (Taken from NYS Testing Draft 7.EE question #2) Which expression below is equivalent
to
+
?
A
(
)
B
(
+
)
C
(
+
)
D
(
+
)
+
2.* Write equivalent expressions for: 3a + 12
3.* An equilateral triangle has a perimeter of 6x + 15. What is the length of each side of the
triangle?
4.** Factor:
−3x + 9
5.** Which students correctly simplified the expressions? Justify your reasoning. Fix all
incorrectly simplified expressions.
Brianda:
=
Sara:
=
Jorge:
=
14
Julia:
=
Trent:
=
6. Factor the following completely: 12x + 8xy
7. The delivery person for a flower shop earns a salary of $40 a day plus $3 for
each delivery he makes. As an expression he would earn (40 + 3d) a day. Write an
expression that would show how much the delivery person will make if he works seven
days delivering flowers.
Expand your expression and explain your work.
Another flower shop offers the delivery person another deal. They will pay him
(90 + 6d) dollars total for 3 days work. Will this job pay more or less than his current
job. Use factoring to compare the two offers.
15
More Combining Like Terms
1.* Suzanne says the two expressions 2(3a – 2) + 4a and 10a – 2 are equivalent. Is she
correct? Explain why or why not.
2. While doing your homework your friend suggests that -6(2x + 9) can be written as -12x + 54. Do you
agree with your friend or suggest a different answer? Explain.
3. In a given Isosceles triangle, the base is found to be half the length of one of the sides. The
expression for the perimeter is given as
. Write the perimeter in simplest form.
4.** Simplify the following linear expression:
+ (
− 7)
5. Find the sum
+
6. Explain why the sum of 6x and 8y is not equal to 14xy
7. Simplify the following expressions. Use number properties and rules to explain each step.
16
A
−
+ [−
+ (
)
B
6x + 4(2 – x) + 3(2x – 1)
8. Can 2.3x + 4.5y + 1.4 be simplified further? Explain why or why not
9. Simplify the following and explain what properties or rules allow you to do this
+ +
+(
)
10. Why must you use the distributive property as your first step in simplifying the following
expression?
2y + 4(3y – 2)
11. Why must you simplify the following expression first before you can factor it?
Explain why then factor it.
2x + 8y + x + 7y
17
12. Find the perimeter of the following triangle
2x + 9
5x − 12
x + 20
13. Simplify
14. Simplify
15. Simplify
16. Is
5x + 8 – (4 + 7x)
(
)
(not the same as #13)
6 – 7(2d – 5) -35
90 + 3(40h – 60) + (-100h) = 30(4h + 3) ?
18
17. Joel has two “x” magnets for his refrigerator and $3. Sarah has three times as much as Joel of both
“x” magnets and money. How much do they have together?
(x + 3) + 3(x + 3) =
Writing Equivalent Expressions
Write the following in two different ways:
1] w + w + 2w + 2w
2] 8p + 2p + 8p + 2p
__________________________________
__________________________________
__________________________________
__________________________________
3] A rectangle is three times as long as it is wide. One way to write an expression to find the perimeter
would be w + w + 3w + 3w. Write the expression in two other ways.
__________________________________
__________________________________
4] Write an equivalent expression for 3(x + 5) – 2. Explain your steps.
5] Ralph reached into his pocket and found some change. He yelled out “Hey! I just found a quarter,
and two dimes, and another quarter, and three more dimes!”. Write this as an algebraic expression in
two different ways.
__________________________________
____________________________________
19
6] Write an equivalent expression for 5(n – 4) – 30. _________________________
7] Which is not an equivalent expression to 7b – 3(b – 4) ?
a) 7b – 3b + 12
b) 4b + 12
c) 4b – 12
d) 4(b + 3)
8] A balloon started from sea level, rose 5 meters, dropped 2 meters, dropped another meter, then rose 8
meters. Write this as a variable expression, then write an equivalent expression.
___________________________________
____________________________
Writing Expressions
Write an expression for each situation:
1. Four less than a number “x”.
____________________________
2. Seven more than a value “n”.
____________________________
3. Five times p pieces of paper.
____________________________
4. D dollars divided 6 ways.
____________________________
5. Twice a value r plus ten.
____________________________
6. Eight less than twice a value g.
_____________________________
7. Nine, less a number k.
_____________________________
8. Fourteen divided by a number s.
_____________________________
9. Three more than four times n.
_____________________________
10. Five times the sum of y and 6.
_____________________________
11. A flower shop owner makes $2 on each flower sold. The shop pays $280 in rent each day
for the store. After the cost of renting the store is deducted, the expression that shows what the
shop owner earns would be (2f – 280) per day. Which expression shows how many dollars the
owner earns in a 7 day week?
20
A
B
C
D
266
14f – 280
14f – 1960
1946f
12. The delivery person for a flower shop earns a salary of $40 a day plus $3 for
each delivery he makes. The expression that shows what he would earn in a day would be
(40 + 3d) .
Write an expression that would show how much the delivery person will make if he works
seven days delivering flowers.
Write an expression for 10 days: _____________
Write an expression for 14 days: _____________
13.* All varieties of a certain brand of cookies are $3.50. A person buys peanut butter cookies
and chocolate chip cookies.
Write an expression that represents the total cost, T, of the cookies if p represents the number
of peanut butter cookies and c represents the number of chocolate chip cookies.
What would this expression look like if the cookies cost $3.75? ________________
14.* Jamie and Ted both get paid an equal hourly wage of $9 per hour. This week, Ted made an
additional $27 dollars in overtime.
Write an expression that represents the weekly wages of both if J = the number of hours that
Jamie worked this week and T = the number of hours Ted worked this week. What is another
way to write this expression?
Write an expression if no one made any additional money from overtime: ___________
15.* Given a square pool as shown in the picture, write four different expressions to find the total
number of tiles in the border. Explain how each of the expressions relates to the diagram
and demonstrate that the expressions are equivalent. Which expression is most useful?
Explain.
21
16. Which of the following expressions has the same meaning as “increase y by 30%”
A
B
C
D
0.30y
0.70y
1.30y
1.70y
17. What does the expression 0.45x represent?
A
B
C
D
increase x by 55%
decrease x by 55%
increase x by 45%
decrease x by 45%
18.** The students in Mr. Sanchez's class are converting distances measured in miles to kilometers.
To estimate the number of kilometers, Abby takes the number of miles, doubles it, then subtracts 20% of
the result. Renato first divides the number of miles by 5, then multiplies the result by 8.
1. Write an algebraic expression for each method.
2. Use your answer to part (a) to decide if the two methods give the same answer.
22
19. (Taken from www.illustrativemathematics.org)
Write an expression for the sequence of operations.
1. Add 3 to x , subtract the result from 1, then double what you have.
2. Add 3 to x , double what you have, then subtract 1 from the result.
20. The Smith family sold their house for d dollars. The real estate agent gets a 2%
commission on the sale price. Write an expression, in simplest form, to represent
how many dollars the Smith family will receive from the sale of their house after
the agent’s commission is deducted from the sale price. Explain your work.
Writing Equations
1.* The youth group is going on a trip to the state fair. The trip costs $52. Included in that price is
$11 for a concert ticket and the cost of 2 passes, one for the rides and one for the game booths.
Each of the passes cost the same price.
Write an equation representing the cost of the trip and determine the price of one pass.
How much would each pass cost if the total $52 was the same, except there were 4 passes? Write an
equation to determine this situation.
2.* Amy had $26 dollars to spend on school supplies. After buying 10 pens, she had $14.30 left. How
much did each pen cost including tax?
Write a sentence to find the cost of one pen. ___________________________
23
What would the equation be if the total cost were $40 and Amy bought 24 pens? __________________
3.* The sum of three consecutive even numbers is 48. What is the smallest of these numbers?
4.** John and his friend have $20 to go to the movies. Tickets are $6.50 each. How much will they
have left for candy? Solve this question in two ways.
5. Jay’s son is not very good at handling money. He likes to spend and he doesn’t like to save. His bank
debit card had a balance of $423.52. After three purchases, each for the same amount, his balance
was −$77.32. What was the amount , p, of each purchase he made. Write an equation and solve it
to find the answer.
6. Sam took a taxi while visiting his daughter in Boston and was charged an initial fee of $4.50 plus $1.75
for every mile I traveled. The ride ended up costing me $53.50. Write an equation and solve it to
find how many miles, m, Sam travelled in the taxi.
24
7. The perimeter of a rectangle is 144 inches. The length of the rectangle is 32 inches. Write an
equation and solve it to find the width, w, of the rectangle.
8. During a recent cookie sale, the cost of a box of cookies was $4.50. The organization gets to keep
20% of that cost, but they then donate $0.20 per box to charity. Write an equation to find the number
of boxes sold if their final profit was $160.
Adding and Subtracting Algebraic Expressions
For each of the following perform the indicated operation:
1. (3x + 6y) + (2x + 8y) = _______________________________
2. (5x – 4) + (–2x + 6) = ________________________________
3. (–8x – 4y) + (–2x + 3y) = _____________________________
4. (4x – 3) + (–6x – 7) = _________________________________
5.
(
)
(
) = ___________________________
25
6. (3x + 6y) – (2x + 8y) = _______________________________
7. (5x – 4) – (–2x + 6) = ________________________________
8. (–8x – 4y) – (–2x + 3y) = _____________________________
9. (4x – 3) – (–6x – 7) = _________________________________
(
10.
) – (
) = ___________________________
Adding / Subtracting Algebraic Expressions
1. (Taken from NYS Testing Draft 7.EE question #1)
When
+
is subtracted from
−
A
−
B
−
C
+
D
+
2. What is the difference when
, the result is
is subtracted from
?
26
3. What is the difference when
is subtracted by
4. What is the sum of
and
?
?
Find the sum of each the following:
(
)
(
(
(
)
)
)
(
(
)
)
(
)
(
)
(
)
(
)
(
)
(
)
27
Find the difference of each of the following:
(
(
(
)
(
)
(
)
(
)
)
)
(
)
(
)
(
)
(
)
(
)
(
)
Multiple Step Word Problems
1. John buys a shirt for “s” dollars. He must pay 6% sales tax on his purchase. Which
expression represents how many dollars he will pay in all, including sales tax?
A
B
C
D
1.6s
1.06s
6s
s + 0.06
2.* Three students conduct the same survey about the number
of hours people sleep at night. The results of the number of people who sleep 8 hours a night are
shown below. In which person’s survey did the most people sleep 8 hours?



Susan reported that 18 of the 48 people she surveyed get 8 hours sleep a night
Kenneth reported that 36% of the people he surveyed get 8 hours sleep a night
Jamal reported that 0.365 of the people he surveyed get 8 hours of sleep a night
28
3.** Malie and her sister won a $45 iTunes gift card. They agree to split the money so that Malie
gets of the value and her sister gets the rest. If songs on iTunes cost $1.29, how many songs
will each sister be able to buy?
4.** Braxton wants to spend his $60 savings on new longboard parts online. He has a promotional code
that he can use for off his cost before shipping or for free shipping. If shipping costs are $1.75 for
each $10 spent, how should he use his promotional code? Justify your answer.
(Taken from www.illustrativemathematics.org)
5. Katie and Margarita have $20.00 each to spend at Students' Choice book store, where all students receive a
20% discount. They both want to purchase a copy of the same book which normally sells for $22.50 plus 10%
sales tax.


To check if she has enough to purchase the book, Katie takes 20% of $22.50 and subtracts that
amount from the normal price. She takes 10% of the discounted selling price and adds it back to
find the purchase amount.
Margarita takes 80% of the normal purchase price and then computes 110% of the reduced price.
Which student is correct? Do they have enough money to purchase the book?
29
6. When working on a report for class, Catrina read that a woman over the age of 40 can lose
approximately 0.16 centimeters of height per year.
1. Catrina's aunt Nancy is 40 years old and is 5 feet 7 inches tall. Assuming her height decreases at
this rate after the age of 40, about how tall will she be at age 65? (Remember that 1 inch = 2.54
centimeters.)
2. Catrina's 90-year-old grandmother is 5 feet 1 inch tall. Assuming her grandmother's height has
also decreased at this rate, about how tall was she at age 40? Explain your reasoning.
7. In Sue’s son’s baby book Sue recorded that he was
recorded that he grew an average of
inches tall on his 2nd birthday. She also
inches each year for the next 12 years. Then he
grew an average of 1 inches each year until his 18th birthday. According to Sue’s
record keeping how tall would her son have been when he was blowing out those birthday
candles at age 18? Is this reasonable? Explain why or why not.
30
8. Students in technology class had the opportunity to create a family name sign. Each student chose
either a ⁄ foot board, a ⁄ foot board, or a ⁄ foot board. If nine students chose the ⁄ ,
seven chose the
to have?
⁄ , and six the
⁄ foot long piece, how much wood did the tech teacher need
One Step Equation Solving
A]
B]
C]
D]
E]
F]
31
G]
J]
H]
(
)
K]
I]
(
)
L]
M]
N]
O]
P] -100p = -1
Q]
R]
Two Step Equation Solving
1. *
− 4 = −16
3. −4y + 9 = −3
2.
(
)
4.
32
5.
6.
7.
8.
9.
10.
33
11.
12.
13.
14.
15.
16.
(
)
Solving Equations by Combing Variables on the Same Side
1.
2.
34
3.
4.
5.
6.
7.
8.
(
)
Solving Equations Using the Distributive Property
1.
(
)
2.
(
)
35
3.
(
)
4.
5.
(
)
6.
7.
(
)
8.
(
))
(
(
)
)
Multiple Step Equation Solving
Solve each of the following showing the proper algebra steps:
1.
3(x – 4) + 15 = 21
2.
2(5x + 4) – 6x = 28
36
3.
8x + 2(x – 3) + 4 = ─22
4.
5.
(x + 12) +
7x ─ 3(2x ─ 4) = ─9
x ─ 2 = ─6
More practice Equation Solving and Checks:
Directions: Answer each question showing each step and a check:
1. –4x = 2.4
Check:
2. 5x – 7 = –37
Check:
37
3.
Check:
4. 8p – 10p = –
Check:
5. 12c + 3c – 14 = 31
Check:
6.
Check:
7. 12c + 3c = 60
Check:
8.
9.
(
)
Check:
9(x + 2) = 47.7
Check:
10. –9(x + 2) = 27
Check:
38
11. 4(x + 2) + 10 = –2
Check:
12. 4x – 9(x + 2) = 2
Check:
13. –2 + 4(3x + 5) = 5
Check:
14. Write an algebraic example of
the Distributive Property
15. Jane is selling 3 paintings at (x + 14) dollars each. If the total of the 3 paintings
comes to $60, find the value of “x”. Use an algebraic equation.
16. A TV is hung on an 8 foot high wall such that the center of the TV rests
5 feet above the floor. If the TV has a frame which is 30 inches high, how
far from the ceiling would the top of the TV frame be?
39
Multiple Step Word Problems
1. (Taken from NYS Testing Draft 7.EE question #3) At a discount furniture store, Chris offered
40
a salesperson $600 for a couch and a chair. The offer includes the 8% sales tax. If the
salesperson accepts the offer, what would be the price of the furniture, to the nearest
dollar, before tax?
A
B
C
D
$552
$556
$592
$648
2. (Taken from NYS Testing Draft 7.EE question #4)
A framed picture 24 inches wide and 28 inches high is shown in the diagram below.
The picture will be hung on a wall where the distance from the floor to ceiling is 8 feet. The
center of the picture must be feet
from the floor. Determine the distance from the ceiling to
the top of the picture frame.
Show your work.
3. Explain why a 20% discount is the same as finding 80% of the cost, c (0.80c)
4.* Amy had $26 dollars to spend on school supplies. After buying 10 pens, she had $14.30 left. How
much did each pen cost including tax?
41
5.* The sum of three consecutive even numbers is 48. What is the smallest of these numbers?
6.** John and his friend have $20 to go to the movies. Tickets are $6.50 each. How much will they
have left for candy?
7. Ken is not very good at handling money. He likes to spend and he doesn’t like to save. His bank
debit card had a balance of $423.52. After three purchases, each for the same amount, his balance
was −$77.32. What was the amount , p, of each purchase he made. Write an equation and solve it
to find the answer.
8. Sue took a taxi while visiting her daughter in Boston and was charged an initial fee of $4.50 plus $1.75
for every mile she traveled. The ride ended up costing her $53.50. Write an equation and solve it to
find how many miles, m, Sue travelled in the taxi.
9. The perimeter of a rectangle is 144 inches. The length of the rectangle is 32 inches. Write an
equation and solve it to find the width, w, of the rectangle.
42
43
Representing Inequalities on the Number Line
44
Write an inequality to represent each graph
45
Solving One Step Inequalities
Solve and Graph
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
14.
15.
13.
(
)
(
)
46
Solving Two Step Inequalities – Solve and graph
1.
3.
2.
(
)
4.
5.
6.
7.
8.
47
Word Problems with Inequalities
A] Jerry needs to buy his sister a birthday gift, and
he also needs to buy a few bottles of soda which
cost $1.28 each. The birthday gift he has in mind
will cost him $43.52. What is the greatest
number of soda bottles Jerry can buy if he has
$60?
B]
Samantha has $56. What is the maximum number
of sunglasses she can buy if they cost $9.45 each?
C]
D]
In a video game, every banana is worth 80 points.
Cara wants to break the high score of 760 points.
What is the least number of bananas she will need
to peel?
A wooden plaque template calls for
inches of
material (including the cut). If Jar Jar has a 3 foot
board, how many plaques can he make?
48
1. Mrs. Pricola’s cell phone company charges her $48.80 a month, plus $0.03 for each text message
she sends. Mrs. Pricola likes to text but she is trying to keep her monthly bill under $75. Write an
inequality an solve it to find the number of text messages, t, that Mrs. Pricola can make and stay
within her budget.
2. Jennifer went to the store with $40 to buy hamburger meat and rolls for a barbecue.
The rolls cost $6. If the hamburger meat is $2.79 a pound, write and solve an inequality
to show how many pounds of hamburger meat Jennifer can buy.
3. Chris earns $45 a day plus $6 for every sale he makes at the store he works in. He needs
to earn at least $125 today to pay back a loan. Write an inequality and solve it to find out
how many sales, s, Chris needs to make.
4.
Jonah gives $50 to his favorite charity plus $0.30 a day to a local charity. Jonah would like to donate at
49
least $75 this years. How many days will he need to give $0.30 to achieve this goal? Write an inequality
and solve it to find out how many days, d, he needs to donate.
5. (Taken from www.illustrativemathematics.org) Fishing Adventures rents small fishing boats to
tourists for day-long fishing trips. Each boat can only carry 1200 pounds of people and gear for safety
reasons. Assume the average weight of a person is 150 pounds. Each group will require 200 lbs of gear
for the boat plus 10 lbs of gear for each person.
A. Create an inequality describing the restrictions on the number of people possible in a rented boat.
Graph the solution set.
B. Several groups of people wish to rent a boat. Group 1 has 4 people. Group 2 has 5 people. Group
3 has 8 people. Which of the groups, if any, can safely rent a boat? What is the maximum number of
people that may rent a boat?
50
6. (Taken from NYS Testing Draft 7.EE question #5)
Mandy’s monthly earnings consist of a fixed salary of $2800 and an 18% commission on all her
monthly sales. To cover her planned expenses, Mandy needs to earn an income of at least $6400
this month.
Part A: Write an inequality that, when solved, will give the amount of sales Mandy needs to cover her
planned expenses.
Answer: ___________________
Part B: Graph the solution of the inequality on the number line.
7. (Taken from NYS Testing Draft question #9)
When John bought his new computer, he purchased an online computer help service. The help service has
a yearly fee of $25.50 and a $10.50 charge for each help session a person uses. If John can only spend $170
for the computer help this year, what is the maximum number of help sessions he can use this year?
51
8.* Florencia has at most $60 to spend on clothes. She wants to buy a pair of jeans for $22 dollars and
spend the rest on t-shirts. Each t-shirt costs $8. Write an inequality for the number of t-shirts she
can purchase.
9.* Steven has $25 dollars to spend. He spent $10.81, including tax, to buy a new DVD. He needs to
save $10.00 but he wants to buy a snack. If peanuts cost $0.38 per package including tax, what is
the maximum number of packages that Steven can buy?
Mixed Review of the Entire Expressions and Equations Domain
1. For each of the following expressions, rewrite in simplest form
A. 6(2x + 3y) – 5x – 2(3y)
B. 4(x – 2y) – 3(y – 2x) + 6x
2. For each of the following expressions, rewrite in factored form
A.
6x – 12y
B.
–4x + 6y + 2
C.
2x(3 – 4y) + 2xy – 4x (Hint: rewrite in simplest form first. Then rewrite in factored form.)
52
3. For each of the following, perform the indicated operation on the given expressions:
A. (7y + 6 ) + (4y – 5)
B. (7y + 6 ) – (4y – 5)
C. 3(x – 4) + 4(2x – 3)
D. 3(x – 4) – 4(2x – 3)
E. The sum of 2x + 6y and – x – y
F. The difference between 2x + 6y and – x – y
G.
4.
2x + 6y subtracted from
When solving the equation
A. Divide both sides by
–x – y
, what might the steps look like?
, then subtract 6 from both sides
53
B. Divide both sides by
, then add 6 to both sides
C. Subtract 6 from both sides, then divide both sides by
D. Add 6 to both sides, then divide both sides by
5.
When solving the equation 2x – 3 + 4x = 15 , what might the steps look like?
A.
B.
C.
D.
Combine the like terms, then add 3 to both sides of the equation, then divide both sides by 6
Combine the like terms, then subtract 3 from both sides of the equation, then divide both sides by 6
Combine the like terms, then add 3 to both sides of the equation, then divide both sides by –2
Combine the like terms, then 3 subract both sides of the equation, then divide both sides by –2
6. When solving the equation
(
) = 15 , what might the steps look like?
A. Multiply both sides by
, then subtract 30 from both sides
B. Divide both sides by
then subtract 30 from both sides
C. Subtract 30 from both sides, then divide both sides by
D. Subtract 30 from both sides, then multiply both sides by
7. Solve each of the following equations by showing the algebra steps:
A.
2x – 2 – 4x = –14
B.
(
) = 15
8. In my last trip to CVS, I bought 4 bags of chips that cost $2.25 each and a magazine. My total before
tax was $13.75. What was the cost of the magazine?
54
9. In Mr. Unson’s last trip to the Gap, he bought a pair of pants for $40 and 3 shirts. He spent a total of $130
for the entire purchase. What was the cost of each of the shirts if they were each the same price?
10. What equation can be written to find the number of textbooks, t, you can buy if the cost of each textbook
is $65 and the math department is giving you m dollars to spend?
A. m = 65 + t
B. t = 65 + m
C. m = 65t
D. t = 65m
11. Mrs. Nuffer paid $165, before tax, for her recent Amazon online order. She ordered v number of $22.50
videos and they charged her a one time charge of $7.50 for shipping. Which of the following equations
could be used to find the number of videos, v, she purchased?
A.
B.
C.
D.
(22.5 + 7.5)v = 165
(22.5 – 7.5)v = 165
22.5v – 7.5 = 165
165 – 7.5 = 22.5v
12. Mr. Welsh’s daughter measured 53 inches at her last checkup at the pediatrician’s office. 6 months
ago when she was measured she was
inches. What was her average growth per month?
13. Greg wants to work overtime to pay for a new cell phone. He needs to earn at least $675 to pay for the
phone he wants. His job will pay him $65 for the first four hours of overtime and $18 for each additional
hour of overtime he works. Write an inequality and solve it to find how many hours of additional overtime
Greg must work to have enough money to buy the phone he wants.
55
14. Part A:
John went to Target and bought his supplies from the school supply list. He bought 4 notebooks for
$4.50 each and 5 1-inch binders at “x” dollars each. The total before tax was $29.25. Write an equation
and solve it to find the cost of each binder.
Part B:
When John made his school supply purchase, Target gave him a $5.00 gift card to use towards his next
purchase. He returned to Target the following week and bought more notebooks for the same price he
paid the week before. If “y” is the number of notebooks he purchased on this second trip and he spent
more than $9, write an inequality and solve it to find the least number of notebooks he bought on the
second trip.
15. A framed picture 36 inches wide and 40 inches high is shown in the diagram below.
56
The picture will be hung on a wall where the distance from the floor to ceiling is 10 feet. The center
of the picture must be 7 feet from the floor. Determine the distance from the ceiling to the top of
the picture frame.
Show your work
16. Mary centered 4 identical mirrors horizontally on a wall that is
inches wide.
Each mirror is 15 inches wide and she left a space of 4 inches between each mirror. To the nearest
inch what is the distance from the end of the last mirror to the end of the wall? (Drawing a picture would
be helpful)
17. The perimeter of a square is 36 inches. What is the length of each side?
57
18. The perimeter of a square is 8x + 16 inches.
In terms of x, what is the length of each side?
19. The perimeter of a square is 20 + 15x inches. In terms of x, what is the length of each side?
20. The perimeter of a square is 12x + 10 inches. In terms of x, what is the length of each side?
21. Mrs. DeMarco took a taxi while visiting NYC. She paid an initial charge of $8 plus $4 per mile.
She paid a total of $32. Write an equation and solve it to find how many miles, m, she
travelled.
22. The Caluri family has only $250 to spend at a water park. They must pay for parking, $15, and the cost
of admission for each family member. Admission is $25.75 per person. Write and solve an equation
that can be used to find the number of family members who can come to the water park.
23. Kyle bought x tee shirts for $10.75 each and y shorts for $24.99 each. The tax was 7.75%.
Part A
Explain why you can express the total cost as
1.0775(10.75x + 24.99y)
58
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Part B
Can you also use
10.75x(0.0775) + 24.99y(0.0775)
Explain why or why not
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
24. What is the product of 4 and ( x + 2.4) in simplest form?
25. Mr Welsh is buying a cover for his rectangular pool.
The total cost of the cover he needs is $975.85.
What’s the cost of the cover per square foot?
26. The following is a diagram of Mrs Wilson’s classroom. “x” represents the width of each
of the doors. The perimeter of the room is 166.8 feet. What’s the width of each door?
59
27. Which of the following is equivalent to
(
A.
)
B.
(
)
C.
(
)
D.
(
)
–6(2y – 4) – 5y + 3
28. Simplify
Review for Quizzes and Tests
Math 7 Warm Up for Quiz
List the properties used to extend each expression:
(
)
(
(
(
)
)
)
60
Math 7 Honors Warm Up for Quiz
List the properties used to extend each expression:
(
)
(
)
(
)
(
) _____________________________________________________________
____________________________________________________________
___________________________________________________________________
61
9
___________________________________________________________________
Combine Like Terms:
Factor Using GCF:
-3 ( 2x
24aw – 8az
+ 5 ) – 6x – 5
Rational or Irrational?
√
Change
0.12777…
̅̅̅̅ into a fraction using the algebra steps:
Math 7 Exp & Equat Test Review
Directions: Choose the best answer to each of the following: Show all work for partial credit.
1. In the following expression, what is the “constant”?
A] -5
2. In the expression
A] coefficient
B] x
C] 21
, the
B] variable
D] No constant
represents the:
C] constant
D] None of these
62
3. Which of the following shows like terms?
A] -4p , -4r
B] -4p , 2p
C] 4p ,
D] None of these
B] 12n
C] -6n
D] None of these
B] 375x
C] -10x
D] None of these
B] 5n – 2
C] -5
D] None of these
B] -10y + 40
C] -10y – 6
D] None of these
4. Simplify:
A] 27n
5. Simplify:
A] 40x
6. Simplify:
(
)
A] 5n – 10
7. Simplify:
(
A] -10y – 40
8.
)
Which of the following illustrates the distributive property?
A]
B]
( )
C]
(
)
D] None of these
63
9.
(
Simplify:
)
A]
B]
(
10. Simplify:
B]
(
)
A]
12. Simplify:
)
(
C]
D] None of these
C]
D] None of these
C]
D] None of these
)
A]
14. Subtract: (
D] None of these
)
B]
)
C]
)
(
A]
13. Add: (
(
B]
(
D] None of these
)
A]
11. Simplify:
C]
B]
)
(
)
64
A]
B]
15. Subtract: (
)
A]
(
C]
D] None of these
C]
D] None of these
)
B]
16. Find the greatest common factor (GCF) of the following:
A] 1
B] 9
C] 6
D] None of these
B] 8c
C] 12cd
D] None of these
B]
C]
17. Find the GCF of the following:
A] 32cd
18. Factor using the GCF:
A]
(
)
(
)
] None of these
Math 7A Expressions & Equations Practice
1. Fill in the blank for each part of the expression:
2. Name the property illustrated by:
_____________________________________
65
3. Write an equation demonstrating the multiplicative inverse property:
______________________________
4. Name the terms in the polynomial: -5n + 2 + 7p.
_____________________________________
Simplify:
5. ____________
6. ____________
7. ____________
8. ____________
(
)
9. ____________
Write “six less than twice a number f” as an expression.
10. ___________
Find the difference when
is subtracted from
11. ___________
Find the difference when
is subtracted by
.
.
Solve for x:
12. __________
13. __________
Solve for x:
14. __________
66
15. __________
16. __________
17. __________
18. __________
–
(
)
19. __________
20. __________
Solve for x:
21. __________
67
22. __________
23. __________
(
)
(
)
Use an equation to solve each of the following:
24. __________
A taxi charges a $3 surcharge, plus $2 per mile. If your bill was $23, how many
miles were you driven?
25. __________
You had $60, then you bought 4 bags (each the same price).
You now have $28 remaining. How much was each bag?
Math 7H Expressions & Equations Practice
1. Fill in the blank for each part of the expression:
68
(
2. Name the property illustrated by:
_____________________________________
)
3. Write an equation demonstrating the multiplicative identity property:
____________________________
4. Name the terms in the polynomial: 8g – 7 – 2x.
_____________________________________
Simplify:
5. ____________
6. ____________
7. ____________
8. ____________
(
)
9. ____________
Write “four less than three times a number n” as an expression.
10. ___________
Find the difference when
is subtracted from
11. ___________
Find the difference when
is subtracted by
.
.
Solve for x:
12. __________
13. __________
Solve for x:
69
14. __________
15. __________
16. __________
17. __________
18. __________
–
(
)
19. __________
20. __________
(
)
70
Solve for x; then graph the solution
21. __________
22. __________
23. __________
(
)
Name the inequality graphed below:
Use an equation to solve each of the following:
24. __________
A taxi charges an $8 surcharge, plus $6 per mile. If your bill was $56, how many
miles were you driven?
25. __________
You had $20 in your pocket, then you bought 5 sodas (each the same price).
You now have $13.25 remaining. How much was each soda?
Review of all Topics So Far
71
Domain 1 – Ratios and Proportional Relationships
MUST SHOW ALL WORK FOR CREDIT
_______1. A recipe calls for
cups of flour for every cup of sugar used.
How many cups of flour are are needed for each cup of sugar used?
_______2. Sara can run
mile in
minutes. What is Sara’s unit rate of speed?
_______3. Ann has a coin that was worth $24 in 2014. It increased in value by 45% in
2015. It then decreased by 20% in 2016. What is the value of Ann’s coin in
in 2016?
_______4. What is the unit rate of speed for the car in the graph?
_______5. Which table or tables show pairs of values that are not in a proportional relationship?
72
A.
x
y
1
1.5
2
3
3
4.5
4
6
x
y
1
5
2
10
3
15
4
20
x
y
1
2
2
4
3
6
4
8
x
y
1
2
3
4
B.
C.
D.
________6. What is the constant of proportionality for the proportional relationship represented
by the equation y = 4.2x
7. For the following situation first write the complex fraction you would use and then
compute the unit rate. Joe traveled
miles per hour.
Complex Fraction: _______________
Unit Rate __________________________
8. Write an equation for the following proportional relationship.
X
1
2
3
4
Y
3.5
7
10.5
14
Equation is _________________________________
73
9. Write an equation for the following proportional relationship.
Equation is _________________________
10. Write the constant of proportionality for the following table.
x
y
1
4.5
2
9
3
13.5
4
18
5
22.5
6
27
The constant is ___________________
11. The graph shows the distance that a bus driver drives in a week.
Does this graph show a proportional relationship? If so, what do the following points
the following points represent in this situation:
(1, 50) and (2, 100) ? Explain your reasoning.
_________________________________________________
_________________________________________________
74
12. The mapping diagram shows that there is a proportional relationship between the
x values and the y values.
Identify the constant of proportionality from the diagram.
Then write an equation to represent the relationship shown
by the diagram. Does your equation also show the constant
of proportionality? Explain
_______________________________________________________________________
________________________________________________________________________
________________________________________________________________________
13. At Best Buy, a TV was marked up 70% of what it cost Best Buy. Best Buy’s cost was
$270. A store employee gets a 30% discount on anything he purchases on the marked
up prices. How much does the employee pay for the TV?
Employee’s Pays ________________
14. Find the Simple Interest on a loan of $65,000 at
% for 4 months. Then find the
balance at the end of the months. Use I = PxRxT (Be careful it says 4 MONTHS)
75
Interest ___________________
Balance ____________________
15. The Game Stop usually sells a game for $27.50 each. They are having a 35% off sale.
Customers must pay an additional 7.75% tax on the sale price of the game. What
will be the total cost of the game, including tax, rounded to the nearest penny?
Final Cost___________________
16. The average temperature in July was 82 degrees. The average temperature in December was
38 degrees. What is the percent of change to the nearest tenth of a percent?
Percent of Change__________________
17.
A store sells books for $10.99. That price includes the $1.42 tax.
What was the store’s percent of tax to the nearest tenth of a percent?
Percent of Tax is _______________
76
Domain #2 - The Number System
SHOW ALL WORK
1. Al wanted to put a railing next to his stairs. The railing needs to be
feet long. If he has
2 pieces of wood that are each
feet long, how much longer must the third piece be?
Write your answer in simplest form.
2. Kim is baking a batch of cookies. She needs
cups of sugar. If she only has
cups, how
much more sugar does Kim need? Write your answer in simplest form.
3. Todd needs to make 3 steps for his deck. One step needs to be
feet long and one step
needs to be
feet long and the third needs to be
feet long. If he has a board of wood
that is 20 feet long, how much more Todd need to make the steps? Write
your answer in simplest form.
4. While doing her math homework, Nicole wrote the following sentence in her notebook:
+(
) = (
)
( )
Which property did she use?
5. Sheila is studying the properties of numbers. Her math teacher wrote this expression on
the board: + 0 = which property does this expression illustrate?
6. Samantha is practicing for the long jump for the upcoming track and field meet. Her first jump
measured
feet. Her second jump was
feet. How much longer was the second
jump?
77
7. Tammy and her two best friends are making two batches of chocolate chip cookies. The
recipe for one batch of cookies is below:
cups of all purpose flour
cup of sugar
cup of packed brown sugar
1 cup (2 sticks) of butter softened
1 teaspoon baking soda
1 teaspoon salt
1 teaspoon vanilla extract
2 large eggs
2 cups (12-oz pkg.) chocolate chips
1 cup chopped nuts (optional)
What is the total amount of flour, sugar, brown sugar and chocolate chips needed to make two
batches of cookies?
8. Last week, Holly bought three five-pound bags of apples and a four-pound bag of cherries.
She made an apple pie and cherry pie for the bake sale. The apple pie called for
pounds
of apples, and the cherry called for
pounds of cherries. How many pounds of apples and
how many pounds of cherries were left after she made the pies?
9. Pet Smart’s brand dog food comes in two sizes:
pounds for the large bag and
for the
smaller size. How many pounds of dog food did Bailey purchase if she bought three large bags and
two small bag of Pet Smart’s dog food?
10. Zach loves winter sports. A typical day at the slopes for him includes practicing snowboard78
ing for
hours, downhill skiing for
hours, and then snow tubing for an hour and a quarter.
How much time does Zach spend on the slopes on a typical day?
11. Joyce is
feet tall. She is at the amusement park with her family and they are about to
ride the newest roller coaster. When they get to the front of the line, there is a sign that
says you must be at least
feet tall to ride the coaster. How much taller than the
required height is Joyce?
12. Bernie spends three hours a day on homework. If he spends of an hour on Science, of
an hour on Social Studies, and of an hour on ELA, how much time does Bernie spend on
math?
13. Use long division to express each of the following fractions as decimals:
A.
B.
C.
14. Cindy wants to make
orders of tacos. Each taco needs
ounces of cheese. How many
ounces of cheese will she need? Express your answer in simplest form.
15. There are 36 students in Ms. Jayne’s third period class.
her class. How many students received an A?
of the students received an A in
79
16. If there are 30 days in June and Steve only worked of the days, how many days did Steve
work? Express your answer in simplest form.
17. Andy pitched
innings a game for 15 games. How many innings did Andy pitch?
18. The area of a rectangle is length x width. Sammy’s swimming pool is shaped like a
rectangle. The length of the pool is
feet. The width is
feet. What is the area
of Sammy’s swimming pool?
19. There are of a pound of grapes left after a picnic. Cody had enough and wants to split what is left
evenly among his three friends. How much will each friend get? Express your answer in simplest
form.
20. Gino has a stick of pepperoni that is
inches long. He wants to cut inch pieces to put
on his large pizza. How many pieces can Gino get from his stick of pepperoni?
21. The area of a rectangle is length x width. What is the length of a rectangular pool table that
has a width of
feet and an area of
square feet? Express your answer in simplest
form.
22. Find the numerical value for each of the following:
A. |
|
B. – |–
|
C. –(–2)
80
D. the additive inverse of
E. the multiplicative inverse of
F. |( (
G. | (
)
)
(– )|
)
|
23. Use the number line to answer the question.
Which statement best models the number line?
A. | | = 3
B. |
|=3
C. | | = –3
D. –3 = 3
24. Write the following subtraction problems as addition problems and find the answer
A. –6 – 5
B. –2 – (–1)
C. 10 – (–10)
25. Ann wrote four statements using absolute value. Which of Ann’s statements is wrong?
A. | | = –8
B. | | = 5
C. |
|=3
D. – | | = –7
81
26. One scuba diver descended 15 meters below the surface of a lake. Another diver
descended 8 meters below the surface. At the same time, a seagull was flying 2 meters
above the lake’s surface, and another seagull was flying 10 meters above the surface.
Which situation has the greatest absolute value in relation to the surface of the lake?
(Hint: The surface of the lake is 0 feet)
A.
B.
C.
D.
The scuba diver that is 15 meters below the lake’s surface
The scuba diver that is 8 meters below the lake’s surface
The seagull that is 2 meters above the lake’s surface
The seagull that is 10 meters above the lake’s surface
27. A model rocket was launched from the ground and shot 150 feet straight up. It then
fell back down to the ground and landed in the same place from which it was launched.
Which expression shows how far the rocket traveled?
A. |
|–|
|
B. |
|–|
|
C. |
|+|
|
D. |
| + (–|
|)
You may use your calculator for this part but must SHOW ALL WORK
28. Sue bought a 3 pounds of flour for cupcake wars at school to share with her friends.
She used
pounds of the flour and divided the remaining equally among her 4 friends.
The expression below represents the number of pounds of flour Sue gave to each of her
friends.
(
)
A. Use the distributive property and rewrite the expression. Explain how
this change would make the expression easier to evaluate.
_____________________________________________________________________
______________________________________________________________________
______________________________________________________________________
B. Evaluate the expression you wrote in Part A. How many pounds of flour did Sue
give to each of her friends?
82
Answer__________
29. Evaluate:
(
)
Show your work
Answer___________
30. John has 9 feet of wood to make new stairs for his deck. He needs
step.
feet for each
A. Set up a complex fraction that represents the number of steps John can make.
Complex Fraction________________
B. Simplify your complex fraction from Part A to solve, and express your answer as a
mixed number.
Mixed Number __________
83
C. Explain what the whole number and fractional parts of the mixed number answer
represent in the context of the situation described.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
31. A number line is shown below.
A. Plot the point
on the given number line
B. What is the opposite of the number
?
______________
C. Using the number line and the definition of additive inverse to explain why the
number 3 and its opposite are additive inverses.
____________________________________________________________________
_____________________________________________________________________
32. Point A is shown on the number line below.
84
What is the additive inverse of the number represented by point A?
33. Evaluate ─12 + 12
Evaluate 25 – 50
34. If a, b, c, and d are non-zero integers, which of the following is equal to
A.
B.
∙
?
C.
35. A number line is shown below.
Which of the following expressions represents the distance between ─2 and 3 on
the number line?
A. |-2+3|
B. |2 - 3|
C. |-3 – (-2)|
36. Divide:
÷
37. Simplify:
÷ 14 SHOW YOUR WORK
D. |-2 -3|
SHOW YOUR WORK
85
38. What is the value of
39. Evaluate
∙(
)
8 ÷ 0 ?
SHOW YOUR WORK
40. Ann simplified the expression shown below as follows.
–15 ─ 25 ─ (– 4) + (─1)
─40 ─ (–4) + (─1)
─36 + (─1)
─37
Step 1
Step 2
Step 3
In which Step was Ann’s mistake? Correct the problem starting on the line where her
mistake occurred and complete the problem to get the correct answer.
41. On the number line shown below, point A has a value of 4
What number must be added to 4 to get a sum of 0 ?
86
42. Evaluate:
–
+ (
) +
HINT: You might want to look at it first and
think about your properties.
MUST SHOW YOUR WORK
43. The energy consumption of an appliance is measured in kilowatt-hours (kWh)
and is the product of the kilowatts per hour the appliance uses and the number
of hours it uses energy. The Unson family’s washer uses 2 kilowatts per hour. If Mr.
Unson runs the washer for
MUST SHOW YOUR WORK.
44.
hours, how much energy will the washer use?
Which of the following situations could be represented by the equation
shown below?
+ (
) = –3
A. The temperature one morning was 4
degrees Celsius. It decreased by 8
degrees Celsius during the day to reach a low of
Celsius.
B. Joe studied for 4 hours, starting at 8:00 PM, so he still needs to study for
3 1/2 more hours.
87
C. Mary spent
dollars on a sandwich. She had 8 dollars, so she
has 3 1/2 dollars left.
D. Louis is 4
miles away from finishing a marathon. He is running at a
speed of 8 miles per hour, so he will finish in 3 1/2 hours.
45. A scuba diver finds some interesting sea creatures at 5 feet below sea level, represented
by the number
–
, and another interesting group of sea creatures at at
feet
below sea level represented by the number
. What is the distance between these
two points below sea level? MUST SHOW YOUR WORK
Domain #3 – Expressions and Equations
1. Distrbute 3(x + 2)
2. Simplify –2(m + 3)
3. Combine like terms to simplify 3x + 5 + 2x
4. Combine like terms to simplify the expression 4m + 2n – m – n
5. Simplify 6t + t
6. Which expression is equivalent to 4x + 3 – 2x + 5
A. 2x + 8
B. 7x – 3
C. 6x + 8
D. 12x + 8
7. Which expression correctly simplifies 4(x – 5)
A. 4x + 5
B. 4x + 20
C. 4x – 5
D. 4x – 20
88
8. Simplify 5x + x
9. Simplify 3r + 2r + r
10. Simplify –2(x + 3)
11. Simplify –7(2x – 4)
12. Simplify 4(4x – 3y + 6)
13. Combine like terms and simplify 6x + 4 + 5x
14. Combine like terms and simplify 16y + 9 + 4 + 4y
15. Combine like terms and simplify 4r – 6 – 2r + 5
16. Combine like terms and simplify 13p + 7 – 14p – 8
17. Which expression is equivalent to 4s + 3 – 3 – s
A. 3s + 6
B. 4s + 0
C. 3s
D. 3s + 1
18. Which expression is equivalent to 6w + 5 – 3w + 7
A. 9w + 2
B. 3w + 12
C. 11w – 4
D. 11w + 4
19. Which expression is equivalent to 3d – d + 4 + d
A. 4d + 4
B. 4d – 4
C. 3d – 4
D. 3d + 4
89
20. Which expression is equivalent to –u + 5 – 3u + 4
A. –4u + 9
B. –3u + 9
C. –2u + 9
D. 2u + 9
21. Simplify 6a + 12b + 5a – 6b + 2
22. Simplify the expression 5g + 5 – 2g + g + 5h
23. Simplify the expression 21q – 4v – 8q + q – v
24. The sum of 2xy and 17xy is
A. 19
B. 19xy
C. 34xy
D. 15xy
25. The difference between 25ab and 37ab is
A. 12ab
B. –12
C. 12
D. –12ab
26. The difference between –7xy and –9xy is
A. –16xy
B. –16
C. 2xy
D. –12xy
27. The sum of –xy and –9xy
28. The sum of –81bc and 7bc
29. The sum of 112cx, 3c, –12cx and 7c
30. 16x + 3xy – 8x – 7xy is equivalent to
90
A. 19xy – 15x
B. –8x – 4xy
C. 8x – 4xy
D. 8x + 4xy
31. –7b – 9ac + 4b + 12ac is equivalent to
A. –3ac + 3b
B. 3ac – 3b
32. Simplify the following expression:
C. ac – b
D. 3b – 3ac
2(xy + 2x) – 3xy
33. Simplify the following expression: –2(3x – 3xy) + xy
34. Simplify the following expression:
x(5y – 3) + 6x – 2xy
35. Simplify the expression 2x(y – 1) – 7xy + 2x
36. For each of the following solve for x by showing the appropriate equation
solving steps:
A.
3x – 4 = 23
B. 5 +
C. –4x – 8 = –16
D. 3.2 + 1.2 = 4.56
E.
F.
91
G. –6x + 21 = 51
H. –x + 25 = 100
I. 6 + 3m = 24
J. x + 2x = 90
K. 5t – 2 – t = 14
L. –3(x – 2) + 1 = 28
M. 6(x – 2) – 4x = 16
N. 9x – 4(x – 3) = 72
O. 15x – 3(3x + 4) = 6
37. Max is saving $10 a month for a summer vacation. If he also has $50 from his
grandmother, how many months will Max need to save to have a total of $200?
Choose the equation that represents this situation.
92
A.
B.
C.
D.
50m + 10 = 200
50m – 10 = 200
10m – 50 = 200
10m + 50 = 200
38. When you solve 4(x + 2) = 22, you must simplify the equation first. What would
the first step look like?
A.
B.
C.
D.
4x + 8 = 22
4x + 2 = 22
x + 8 = 22
x + 2 = 22
39. When solving equations, sometimes you have to simplify the equation first.
When solving 3x + 2 + x – 6 = 25, what might your first step look like?
A. 3x + 8 = 25
B. 3x – 4 = 25
C. 4x + 8 = 25
D. 4x – 4 = 25
40. James bought a sweater that cost $15.99 and some socks that cost $0.99 per
pair. The total cost was $21.93. Write an equation and solve it to find how many
pairs of socks James bought.
41. Yesterday, Mary mailed her wedding invitations. Today she mailed 72 more
invitations than she did yesterday. If Mary mailed 200 invitations in all, how many
invitations did she mail yesterday?
93
42. John bought 3 cars in the last year for a total of $30,000. He paid the same amount
for each of the first 2 cars, and got the 3rd one for twice the amount of the 1st car.
Which equation can be used to find out how much John paid for each car?
A.
B.
C.
D.
x(30,000) = 2x
x + 2x = 30,000
30,000x = x + 2x
x + x + 2x = 30,000
43. There are 3 flights that depart from New York and arrive in London. Combined, the
3 planes hold 1,200 people. If there are 405 seats in the first plane, and 345 more
seats in the third plane than in the second, how many seats are in the second
plane?
A.
B.
C.
D.
405 + 1200 + x + x = 345
405 + x + x + 345 = 1200
405 + x + 345X = 1200
405(2x) + 345 = 1200
44. If you add 125 to of a number, the result is 450. Which equation would
help find the original number?
A.
(
)
B.
C.
D.
45. Yesterday Bill bought some bottles of soda for a party he is hosting. Today he
bought 8 times as many bottles of soda as he did yesterday. Bill has 27 bottles
94
in all. Which equation would help you to find out how many bottles of soda Bill
bought yesterday?
A.
B.
C.
D.
x + 8x = 27
8x – x = 27
8x(x) = 27
8x ÷ x 27
46. Which inequality represents the translation of the following sentence?
“Twice a number increased by four is more than 7”
A.
B.
C.
D.
2x + 4 < 7
2x – 4 > 7
2x + 4 > 7
2x + 4 ≤ 7
47. Which inequality represents the translation of the following sentence?
“Half a number is less than four more than the same number”
A.
C. 2x > x + 4
B.
D.
48. Which inequality represents the translation of the following sentence? “The
product of a number and three is less than or equal to that number decreased by five”
A.
B.
C.
D.
3x ≤
3x ≥
3+x
3x ≤
5–x
x–5
≤ x–5
x–5
49. Which inequality represents the translation of the following sentence?
“Four more than two times a number is less than or equal to five”
A. 4 + 2x > 5
B. 2x + 4 < 5
95
C. 4 + 2x ≥ 5
D. 2x + 4 ≤ 5
50. Which inequality represents the translation of the following sentence?
“Twelve less than a number is greater than that same number divided by two”
A. x – 12 ≥ 2x
B. 12 – x ≥
C. x – 12 >
D. x – 12 <
51. Which inequality represents the translation of the following sentence?
“The quotient of a number and 5 is less than or equal to that number increased by 1”
A. 5x < x + 1
B.
C.
D.
52. Vincent has cut three pieces of rope to complete a science project. Two pieces are
of equal length. The third piece is one-quarter the length of each of the others.
He cut the three pieces from a rope 54 meters long without any rope left over.
Find the number of meters in each piece.
53. A restaurant sells large and small submarine sandwiches. Rolls for the sandwiches
are ordered from a baker. The roll for the large sandwich costs $0.25 and the roll
96
for a small sandwich costs $0.15. Melissa, the manager of the restaurant, ordered
130 more large rolls than small rolls. What was the greatest number of large rolls
she received if she spent less than $63.
54. Two numbers are in the ratio 5:6. If the sum of the numbers is 66, find the value
of the larger number.
A. 6
B. 24
C. 30
D. 36
55. At a concert, $720 was collected for hot dogs, hamburgers, and soft drinks. All
three items sold for $1.00 each. Twice as many hot dogs were sold as hamburgers.
Three times as many soft drinks were sold as hamburgers. The number of soft
drinks sold was
A. 120
B. 240
C. 360
D. 480
56. A boy got 50% of the questions on a test correct. If he had 10 questions correct
out of the first 12, and of the remaining questions correct, how many
questions were on the test?
A.
B.
C.
D.
16
24
26
28
57. A doughnut shop charges $0.70 for each doughnut and $0.30 for a carryout box.
Shirley has $5.00 to spend. At most, how many doughnuts can she buy if she also
97
wants them in one carryout box?
58. There are 28 students in a mathematics class. If
the guidance office,
of the students are called to
of the remaining students are called to the nurse, and
finally, of those left go to the library, how many students remain in the
classroom?
59. Peter begins his kindergarten year able to spell 10 words. He is going to learn to
spell 2 new words every day. Which inequality can be used to determine how many
days, d, it takes Peter to be able to spell at least 75 words?
A.
B.
C.
D.
10 + 2d ≥ 75
10 + 2d ≤ 75
2 + 10d ≥ 75
2 + 10d ≤ 75
60. Tamara has a cell phone plan that charges $0.07 per minute plus a monthly fee of
$19.00. She budgets $29.50 per month for total cell phone expenses without
taxes. What is the maximum number of minutes Tamara could use her phone
each month in order to stay within her budget?
A.
B.
C.
D.
150
271
421
692
98
61. The table below represents the number of hours a student worked and the
amount of money the student earned.
Number of Hours
(h)
8
15
19
30
Dollars Earned
(d)
$50.00
$93.75
$118.75
$187.50
Which equation correctly represents the number of dollars, d, earned in terms
of the number of hours, h, worked?
A.
B.
C.
D.
h = 6.25d
d = 6.25h
h = 50 + 6.25d
d = 50 + 6.25h
62. An online music club has a one-time registration fee of $13.95 and charges $0.49 to
buy each song. If Emma has $50.00 to join the club and buy songs, what is the
maximum number of songs she can buy?
A.
B.
C.
D.
73
74
130
131
63. The ninth grade class at a local high school needs to purchase a park permit for
$250.00 for their upcoming class picnic. Each ninth grader attending the picnic
pays $0.75. Each guest pays $1.25. If 200 ninth graders attend the picnic, which
inequality can be used to determine the number of guests, x, needed to cover the
cost of the permit?
A.
B.
C.
D.
0.75x – (1.25)(200)
0.75x + (1.25)(200)
(0.75)(200) – 1.25x
(0.75)(200) + 1.25x
≥
≥
≥
≥
250.00
250.00
250.00
250.00
99
DIRECTIONS: Find the mistake in each question; indicate where the first mistake occurs.
Describe the mistake and the correction.
Expand each expression
#64
3(x + 9)
3x + 9
#65
6(x + 2)
6x + 6(2)
6x + 12
18x
#66
5(x – 6)
5(-5)
-25
100
Factor each expression
#67
4n + 8
4(n + 2)
4(3)
12
#68
20n – 4ny
2n(10 – 2y)
#69
-18n + 8
2(-9n – 4)
Simplify
#70
#71
#72
Enrichment
ALL OF THE FOLLOWING ARE TAKEN FROM NORTH CAROLINA’S CCSS (They had a footnote acknowledging
that some of the problems and graphics were taken from the Arizona Department of Education)
8.EE.1
1.
101
2.
3.
4.
5. (
)(
6. (
)
7.
(
)
)
( ) (
)
8.EE.2
1. difference between (
2. Understanding ( )
3. Solve
=
4. Solve
= 27
5. Solve
=
)
and understanding that √
= ±4
and √
6. What is the side length of a square with an area of 49
8.EE.3
1. Write 75,000,000,000 in scientific notation
2. Write 0.0000429 in scientific notation
3. Express 2.45 x
in standard form
102
4. How much larger is 6 x
compared to 2 x
5. Which is the larger value: 2 x
or 9 x
8.EE.4
1. Problems using scientific calculators using E or EE (scientific notation), *(multiplication), and
^(exponent) symbols
2. In July 2010 there were approximately 500 million facebook users. In July 2011 there were
approximately 750 million facebook users. How many more users were there in 2011. Write
your answer in scientific notation.
103
3. (6.45 x
)(3.2 x
)
4.
5. (0.0025)(5.2 x
)
6. The speed of light is 3 x
meters/second. If the sun is 1.5 x
meters from earth, how
many seconds does it take light to reach the earth? Express your answer in scientific
notation.
8.EE.5
1. Compare the scenarios to determine which represents a greater speed. Explain your
choice including a written description of each scenario. Be sure to include the unit rates
in your explanation.
104
Scenario 1:
Scenario 2:
Traveling Time
y = 55x
(shows a graph with time
as the horizontal axis,
distance (miles) as the
vertical axis and the line
drawn includes the points (1,60)
and (4, 240)
x is time in hours
y is distance in miles
8.EE.6
1. (Shows coordinate plane with two triangles shaded in) The triangle between A and B has
a vertical height of 2 and a horizontal length of 3. The triangle between B and C has a
vertical height of 4 and a horizontal length of 6. The simplified ratio of the vertical height to
the horizontal length of both triangles is 2 to 3, which also represents a slope of
for
the line, indicating that the triangles are similar.
2. Given an equation in slope-intercept form, students graph the line
a.
b.
c.
d.
3. Students write equations in the form y = mx for lines going through the origin and recognize
that m is the slope of the line
See hand out
4. Write an equation of a line that is graphed that goes through the origin.
Write an equation of a line that is graphed that does not go through the origin
understanding y = mx + b
105
8.EE.7
1. Solve 10x – 23 = 29 – 3x
2. Equations that have no solution because the variables cancel out:
–x + 7 – 6x = 19 – 7x
3. Equations with infinitely many solutions when both sides are the same:
(
)
(
)
4. Two more than a certain number is 15 less than twice the number. Find the number.
106
8.EE.8
1. Plant A and Plant B are on different watering schedules. This affects their rate of growth.
Compare the growth of the two plants to determine when their heights will be the same.
(Given a table, then based on the table they make a graph. Then they write an equation
for each representing the growth rate of Plant A and the growth rate of Plant B)
Determine at what week will the plants have the same height based on your equations.
Write and solve an equation to find the week.
2. Victor is half as old as Maria. The sum of their ages is 54. How old is Victor?
Multiplication and Division Property of Inequalities
107
If you multiply or divide each side of an inequality by a positive number,
leave the inequality symbol unchanged.
If you multiply or divide each side of an inequality by a negative number,
reverse the inequality symbol.
Example: When you multiply or divide by
a negative in the last step
you have to switch the sign around
-4x > 20
-4x > 20
-4 -4
x < -5
When you multiply or divide by
a negative in the last step you have to
switch the sign around.
x<9
-3
-3x < 9(-3)
-3
x > -27
**But 2x > -10 you are not going to switch the sign around
Practice: Solve and graph each of the following
108
1. –3x + 9 ≥ –21
2. x + 15 > 23
-8
4. 55 > –7x + 6
5. –25 + x ≤ 50
2
7. 10 + x > 12
–9
8. –6x – 18 ≥ 36
3. 2x + 5 > -49
6. x – 22 < 48
2
9. –2x + 12 > 8
109
Multiple Step Equation Solving with Variables on Both Sides
1.
12x - 3 = 8x + 21
2. 5(2x + 3) + 2x = x + 48
3.
3(2x + 4) = 2(2x - 20)
4.
4(x – 2) = 3x + 1
25y + 10 - 15y + 20 = 100
5.
10(y + 6) + 3y = 86
6.
7.
14x + 33 - 3x - 11 = 33
8. 3(y + 5) + 2y + 10 = 50
9. 3(y – 2) + 6 = 90
10. 2y + 17 + 3 + 3y = 100
110
11. 4(x + 3) = 2x - 24
13.
12x – 4 = 2x + 6
12. 2x + 3(x + 1) = x + 31
14.
3(x + 2) + 5 = 2x + 44
15. 2(x – 5) + 8 = 16
16. 4(x – 2) = 3(x + 4)
17.
10y + 3 = 6y + 27
18.
-2y + 6 = 4y - 36
19.
x(6 – 3) + 12 = x + 20
20.
2(y + 2) = 5(y – 4)
111